There was nothing about him wanting to meet Loki - the opposite, he was trying to stop him for 5 out of 6 episodes. There was no hint that he was growing tired of watching the timelines.
If you want a superhero show that is about the villain, DC's really the place to go. The show is called "Loki" not "The Man At the End of Time," so it is about Loki. Marvel shows are about the heroes. The villains are there to be defeated, not to have their psychology explored. (Spider-Man movies are somewhat the exception to this tendency)
About how Kang Who Remains wanted to stop Loki - he actually didn't want to stop Loki. But Loki, a generally distrustful person, can't just be called in and given a job offer. If anything is too easy, Loki gets suspicious, and backs away - because if Loki were to offer you something easily, it would be because Loki is setting you up for betrayal. Kang used this baroque approach of feigned resistance because the way to make Loki show up somewhere is to tell Loki he cannot go there.